Some Applications Of Operator Theory In Quantum Information And Non-commutative Computation

Operator matrices, numerical ranges of operators, spectral theory of operators and commutaors are some heatly discussed topics in operator theory. The research of these subjects has been related to algebra, matrix analysis, non-commutative geometry, non-commutative computation and quantum computation, etc. The dissertation is related to some applications of operator theory in quantum information and non-commutative computation. The research methods mainly focus on techniques of block operator and spectral theory. The dissertation, which can be divided into five chapters, mainly studies the ranges of upper triangular operator matrices, numerical ranges of operators, spectral characterizations of generalized quantum gate and infimum of two self-adjoint operators, the maximum norm of commutators of positive contractions, and oblique projectors.In Chapter 1, by using the technique of block operator matrices, for the operator matrix in which operators A and B are on the diagonal, under the four cases that R(A) and R(B) are closed or both are not closed, or one of them is closed and the other is not closed, the closedness of R(MC) is studied. Combining the obtained con-clusions, the sufficient and necessary conditions under which the upper triangular operator matrix is Kato non-singular are given.In Chapter 2, for the fundamental theorem concerning the numerical ranges of self-adjoint operators which was used by S. Gudder and G. Nagy when studying the theory of sequential quantum measurements, the restriction on operators which are self-adjoint is removed and it is generalized to general bounded linear operators. The open problem con-cerning the minimum modulus and inner numerical radius of matrix, which was raised by P. J. Psarrakos and M. J. Tsatsomeros, has been solved completely. The corresponding conclusion is also proved to be correct for operators on an infinite dimensional Hilbert space. Finally, by using the technique of block operator matrices, the research reveals the relations between the corner of numerical range and the point spectrum together with the reduced approximation point spectrum.In Chapter 3, several problems concerning the generalized quantum gate and self-adjoint operators with respect to the logic order are studied, which were introduced by G. L. Long and S. Gudder when studying the principle of duality quantum computers and the quantum measurements, respectively. By using the spectral theory of linear operators and the technique of block operator matrices, the research puts forward spectral representations of generalized quantum gate and infimum of two self-adjoint operators with respect to the logic order.In Chapter 4, by using the spectral theory of linear operators, in the non-commutative computation setting, the problem concerning commutators of two positive contract opera-tors which was raised by M. D. Choi is studied. The sufficient and necessary conditions under which the maximum norm attainability of positive contract commutators are estab-lished.In Chapter 5, starting with the geometry structure of oblique projectors, the oper-ator matrix representation of an oblique projector for complementary subspaces is given. For operators acting on an infinite dimensional Hilbert space, some characterizations of the range and the null space of an oblique projector U(VU)(?)V are obtained, and the representation of a general oblique projector is also given.The results from the dissertation consist of the following seven statements.1. By using the block operator matrices technique, the sufficient and necessary con-ditions under which the range closedness and the Kato non-singularity of upper triangular operator matrices under several different conditions are given.2. We prove that for A, B, C, which are bounded linear operators on a Hilbert space H, if (Ax,x)(Bx,x)= (Cx,x) for all x∈H with‖x‖= 1, then at least one of A and B is a scalar multiple of the identity, i.e. there exists a complex number c∈C such that A=cI or B=cI.3. For every bounded linear operator A on a Hilbert space H, If 0 (?) W (A),thenδ(A)≥(?)(A).4. By using the spectral theory of linear operators, a sufficient and necessary condition under which a contract operator is a generalized quantum gate is established.5. A spectral representation of the infimum of two self-adjoint operators with respect to the logic order is given.6. Some conditions under which the maximum norm attainability of commutators of two positive contractions are attained.7. Some representations of the range and the null space of an oblique projector U(VU)(?)V are given, and the representation of a general oblique projector acting on an infinite dimensional Hilbert space is also established.